First if you can, try to use different cable for monitor. It could be that cable is damaged. I've had seen problems such as yours on with damaged DVI->VGA adapters. If the cables are good, try with DVI->VGA adapter. It may help you if the VGA port on the card is damaged. If the monitors themselves have DVI ports, buy appropriate cable and do use them!
Also, check settings and make sure that they are OK. Try with some recent GNU/Linux LiveCD. If it works fine, try with driver reinstallation.
If all fails, post your graphics card's chipset and manufacturer. There might be a way to save it by baking it, if it's using certain chipset; but it's a drastic measure which should be applied to cards which would otherwise go to trash-can.
Oh and "motherboard's inbuilt graphics output" is usually called integrated video card.
EDIT:
You provided new information, so here's my response: clean your graphics card's cooling system! I've seen people talk on INTERNET that on older graphics cards, the solder ball grid array which connects the card's processor to circuit board can start to crack because the solder balls expand due to effects of heat. After the card cools down, they shrink. I believe that the best way to solve this is to wait until the card becomes unusable (you can prolong the time by providing better cooling) and then research card baking.
If you manage to set correct temperature and properly heat up and later cool down the card, the solder balls may melt in a correct way and re-solder the processor correctly. There's some discussion of that on its dedicated question here. Also take a look here for some m0re information.
On the other hand, it could be that the processor itself is cooked. Graphics cards of nVidia's 8000 series are especially susceptible to that problem because of low quality of raw materials used in manufacture. If it's this problem, then I'm afraid that there is no way to actually repair the card. One day it will fail and you'd have to obtain a new card.
